Lieb Schultz Mattis-Type Theorems and Other Non-perturbative Results for Strongly Correlated Systems with Conserved Dipole Moments
LLieb Schultz Mattis-Type Theorems and Other Non-perturbative Results for StronglyCorrelated Systems with Conserved Dipole Moments
Oleg Dubinkin, ∗ Julian May-Mann, † and Taylor L. Hughes ‡ Department of Physics and Institute for Condensed Matter Theory,University of Illinois at Urbana-Champaign, IL 61801, USA
Non-perturbative constraints on many body physics–such as the famous Lieb-Schultz-Mattistheorem–are valuable tools for studying strongly correlated systems. To this end, we present anumber of non-perturbative results that constrain the low-energy physics of systems having con-served dipole moments. We find that for these systems, a unique translationally invariant gappedground state is only possible if the polarization of the system is integer. Furthermore, if a latticesystem also has U (1) subsystem charge conservation symmetry, a unique gapped ground state isonly possible if the particle filling along these subsystems is integer. We also apply these methods tospin systems, and determine criteria for the existence of a new type of magnetic response plateau inthe presence of a non-uniform magnetic field. Finally, we formulate a version of Luttinger’s theoremfor 1D systems consisting of dipoles. I. INTRODUCTION
Strongly correlated quantum systems are often thesource of striking phenomena, yet they remain one of themost challenging to analyze. If perturbative approachesfail, one’s only recourse is numerical simulation unlessnon-perturbative methods or results can be applied. Oneof the most celebrated non-perturbative results is theLieb-Schultz-Mattis (LSM) theorem, which from very lit-tle information, i.e., the number of spin-1 / . This result hashad wide-ranging applicability in quantum systems, andhas been extended to higher dimensions .A key feature of the proof of the LSM theorem is a twistoperator that slowly rotates the spins across a spin-chain.In the work of Oshikawa , which applied and generalizedthe LSM result to lattice systems with conserved particlenumber, a related twist operator is used U X = exp (cid:34) πi ˆ XL x (cid:35) , (1)where ˆ X = (cid:80) x x ˆ n x is the many-body position operator.Indeed, Oshikawa and collaborators also used this opera-tor to provide a non-perturbative understanding of Lut-tinger’s theorem , determine the Fermi surface proper-ties of the Kondo lattice , and more recently to cal-culate filling-enforced constraints on the quantum Hallconductivity in lattice systems . Remarkably, this op-erator U X has had parallel uses in the theory of elec-tronic polarization where it was introduced by Resta .In this context, the complex phase of the ground stateexpectation value of U X is determined by the electronicpolarization , and the magnitude is determined by theelectron localization length .In recent work, new twist operators have been pro-posed whose ground state expectation values can be usedto calculate higher multipole moments . The sim- plest multipole generalization of U X is the operator U XY = exp (cid:34) πi (cid:100) XYL x L y (cid:35) (2)where (cid:100) XY = (cid:80) x xy ˆ n x is the many-body quadrupoleoperator. In light of this development it is naturalto use these multipole operators to try to derive non-perturbative results analogous to the previous work onthe LSM theorem , Luttinger’s theorem , magnetiza-tion plateaus , and filling-enforced Hall conductivityconstraints . In this article we focus on higher mul-tipole generalizations of some of these results, derivedthrough the application of operators related to U XY ineach context. Our goal is to recast the original resultsthat apply to particles/charges to apply to dipoles. No-tably, we study generalizations of the LSM theorem forsystems that conserve dipole moments (Section II), andthen apply these results to study magnetization (gradi-ent) plateaus in spin systems (Section III), and an exten-sion of Luttinger’s theorem to dipole conserving systems(Section IV). We also extend our results on the LSM-type theorems to systems with U (1) subsystem symme-try, e.g., symmetries enforcing charge conservation alongrows or columns in 2D . These results are applica-ble to some fracton systems, and it is possible they mayeventually be adapted in some form to systems with bro-ken subsystem symmetry (and broken microscopicdipole conservation), e.g., higher order multipole bandinsulators of fermions or bosons , though we leavethose developments to future work. II. LSM-TYPE THEOREMS FOR DIPOLECONSERVING SYSTEMS
Let us briefly recount the concept behind Oshikawa’sproof of the LSM theorem that we wish to generalize.His starting point was a system with U (1) particle conser-vation, and he utilized a twist operator U X that achieves a r X i v : . [ c ond - m a t . s t r- e l ] J a n two different things simultaneously: when applied to theground state of a periodic insulating system, it extractsthe charge polarization of the ground state, and it per-forms a large gauge transformation on this state, i.e.,it effectively inserts or removes a single unit of mag-netic flux through the periodic loop running along theˆ x -direction. To see how this works, consider a nearest-neighbor lattice model of free fermions in 1D with N x sites along ˆ x : H = − t (cid:80) N x j =1 ( c † j +1 c j + h.c. ). Applying U X to the Hamiltonian modifies every fermionic hop-ping term by attaching a phase factor of e πi/N x , which isequivalent to introducing an external gauge field A x witha 2 π circulation along the loop going around the periodicˆ x -direction. If U X is applied to the ground state | Ψ (cid:105) ofthis free-fermion system, for example, then, in the ther-modynamic limit, a set of filled bands will return backto themselves up to a phase proportional to the polariza-tion, while a partially filled band will change momentumand be orthogonal to the original state. This is a verysimple application of Oshikawa’s results that indicatesthat partially filled bands cannot support an excitationgap since the energy of the state U x | Ψ (cid:105) will nominallyapproach that of | Ψ (cid:105) itself in the thermodynamic limit,and the two states are orthogonal if there are partiallyfilled bands because they have different momentum quan-tum numbers. A. Dipole Conserving Systems
To make progress toward an LSM-type theorem for sys-tems with charge and dipole conservation, let us considera system defined on a rectangular periodic L x × L y = N x a × N y a, lattice where a is the lattice constant. Wewill work with Hamiltonians H that are translationallyinvariant and conserve both global particle and dipolenumber. Hence, the Hamiltonian is invariant when thecharged operators are changed by constant phase trans-formations e iα , and phase transformations with linear co-ordinate dependence e i α · x , respectively. The latter con-dition also automatically implies that H commutes withthe twist operators U X and U Y , i.e., the total polariza-tion is a fixed quantum number for eigenstates of H. Wenote that it has been shown that systems with dipoleconservation can be coupled to a rank-2 gauge field A ij .The rank-2 gauge field transforms as A ij → A ij + ∂ i ∂ j λ under a gauge transformation, where λ is a generic func-tion.Similar to Oshikawa’s work, we will be consideringthe action of generalized U XY twist operators on theground states of insulating systems: U XY ( α ) = e iα (cid:100) XY .For α = πL x L y , and for systems with open boundaries,this operator was shown to be successful in extractingthe quadrupolar polarization and, when applied tosystems that explicitly conserve dipole moment, it intro-duces a constant rank-2 gauge field A xy across the lattice.However, if periodic boundary conditions are introduced, U XY ( πL x L y ) exhibits problematic behavior. For instance,while the complex phase of its ground-state expectationvalue (cid:104) U XY ( πL x L y ) (cid:105) , computed for the ground states offree-fermion tight-binding models, correctly captures thequadrupolar polarization, the absolute value of the ex-pectation value vanishes when the thermodynamic limitis taken because of fluctuations of the dipole moments.Even if we mitigate this issue by restricting ourselves toHamiltonians with manifest dipole conservation, as weshall do in this article, the dipole-conserving terms in theHamiltonian that cross the periodic boundary pick up anadditional phase factor under the action of U XY ( πL x L y ) . To see this, one can act on dipole-conserving terms in theHamiltonian, e.g., ring-exchange terms U − XY ( α ) (cid:16) c † x,y c x + a,y c † x + a,y + a c x,y + a (cid:17) U XY ( α )=e iα c † x,y c x + a,y c † x + a,y + a c x,y + a . For terms that cross a periodic boundary additionalphase factors are generated, e.g., e iαL x a or e iαL y a . In-deed, one can check that in order to have U XY ( α ) inserta constant rank-2 gauge field A xy for a system with peri-odic boundary conditions then αL x a = 2 π Z and αL y a =2 π Z . Hence, in order to be consistent with periodicboundary conditions, we will choose α = πa gcd( N x ,N y ) where a is the lattice constant in the x and y directions.To proceed from this setup, let us consider adiabati-cally turning on a constant gauge field configuration of A xy over a time T having the form A xy = 2 πa gcd( N x , N y ) tT . (3)Let us label the Hamiltonian as a function of time as H ( t ) , and its instantaneous ground state as | Ψ( t ) (cid:105) . Sincethe initial system is translationally invariant, | Ψ(0) (cid:105) is aneigenstate of the many-body translation operators T x and T y that send each particle coordinate ( x, y ) to ( x + a, y )or ( x, y + a ) respectively. We will take the T x eigen-value to be e iP x and the T y eigenvalue to be e iP y . Since H ( t ) is translationally invariant at all times, | Ψ( t ) (cid:105) willremain an eigenstate of T x and T y with the same eigen-values e iP x and e iP y at all times. Similarly, since H ( t )commutes with both U X and U Y at all times, | Ψ( t ) (cid:105) willalso remain an eigenstate of U X and U Y with eigenval-ues e πiX /L x and e πiY /L y at all times. At t = T ,we have A xy = πa gcd( N x ,N y ) , which is equivalent to a(large) gauge transformation that can be removed byapplying the many-body twist operator U XY ( α g ) where α g = πa gcd( N x ,N y ) . As a result H (0) = U − XY ( α g ) H ( T ) U XY ( α g ) , (4)and so U − XY ( α g ) | Ψ( T ) (cid:105) is also a ground state of H (0).The next key step for an LSM-type theorem is to de-termine if U − XY ( α g ) | Ψ( T ) (cid:105) and | Ψ(0) (cid:105) are orthogonal. Todo this, we will compare the eigenvalues of the transla-tion operator for U − XY ( α g ) | Ψ( T ) (cid:105) and | Ψ(0) (cid:105) . A simplecalculation shows that T x U − XY ( α g ) | Ψ( T ) (cid:105) = exp (cid:34) πi ˆ Ya gcd( N x , N y ) (cid:35) e iP x U − XY ( α g ) | Ψ( T ) (cid:105) . (5)We already know that T x | Ψ(0) (cid:105) = e iP x | Ψ(0) (cid:105) , so theground state is unique only if exp (cid:104) πi ˆ Ya gcd( N x ,N y ) (cid:105) | Ψ( T ) (cid:105) = | Ψ( T ) (cid:105) . If we define n = L y / ( a gcd( N x , N y )), thenexp (cid:104) πi ˆ Ya gcd( N x ,N y ) (cid:105) = ( U Y ) n , andexp (cid:34) πi ˆ Ya gcd( N x , N y ) (cid:35) | Ψ( T ) (cid:105) = e πinY /L y | Ψ( T ) (cid:105) , (6)where we have used that U Y | Ψ( T ) (cid:105) = e πiY /L y | Ψ( T ) (cid:105) .Since ( U Y ) N y = 1, e πiY /L y must be an N th y root ofunity, and Y /L y ≡ πi log( e πiY /L y ) must be a rationalnumber. We can define Y /L y = p/q , where p and q areco-prime. So U − XY ( α g ) | Ψ( T ) (cid:105) is an eigenstate of T x witheigenvalue exp( iP x + 2 πinp/q ). If n and q are co-prime U − XY ( α g ) | Ψ( T ) (cid:105) must be orthogonal to | Ψ(0) (cid:105) , and thusthe system is either gapless, or there must be at least q degenerate ground states if the system is gapped.This statement relies on n and q being co-prime, but if n is an integer multiple of q nothing can be said about thedegeneracy of the ground state. A similar issue was re-marked upon in Oshikawa’s proof of the LSM theorem .Here, we can avoid this issue by requiring that the ther-modynamic limit is taken such that N x = N y = N . Sinceground state properties should be independent of howthe thermodynamic limit is taken, we can assume thatthe the thermodynamic limit is taken in this way. Fromthis we can conclude that the ground state of the sys-tem is unique only if e πiY /Na = 1. Using the samelogic, i.e., by acting with the translation operator T y , wecan also conclude that the ground state is unique only ife πiX /Na = 1 as well. In other words, for the groundstate to be unique, we must require that the compo-nents of the polarization in both directions vanish upto a polarization quantum. The bulk of a locally electri-cally neutral system must carry a uniform polarization,which allows us to relate a microscopic dipole momentstretching between the pair of neighboring unit cells to amacroscopic polarization of the system picked up by thephase of the unitary operator e πi ˆ X j /N j a . Therefore, wesee that the pair of conditions e πiY /Na = e πiX /Na = 1is equivalent to requiring that the microscopic dipole mo-ments stretching between every pair of neighboring unitcells must be an integer times ea . Thus we conclude thatfilling factor for x and y dipoles must be an integer, anal-ogous to the requirement that the charge filling factor beinteger in the conventional LSM theorem. B. Subsystem Symmetric Systems
We can construct a stronger version of the above dipoleLSM theorem for Hamiltonians with dipole conservationarising from U (1) subsystem symmetry. Let us first pro-vide a brief background discussion on subsystem sym-metries. To give an example, consider a 2-dimensional L x × L y rectangular lattice. The subsystem symmetry op-erator corresponding to U (1) charge conservation alonga single column with x = x is given by: U ,x ( α ) = exp iα L y (cid:88) y =1 ˆ n x ,y . (7)Such operators rotate the phase of all electrons along asingle column in the lattice. In other words, U ,x can bethought of as a restriction of the global U (1) symmetryoperator U ( α ) = exp (cid:16) iα (cid:80) L x ,L y x,y =1 ˆ n x,y (cid:17) to a particularsubsystem. Similarly, we can define subsystem symme-tries U ,y that impose charge conservation along everysingle row y of the lattice. For the purposes of our work,by an n -dimensional subsystem in a d -dimensional Bra-vais lattice, we will understand an n -dimensional latticesubspace spanned by any n linearly independent latticebasis vectors. Now, taking a collection of such “parallel”subspaces that cover the entire lattice, we can impose U (1) charge conservation along each of the subspaces in-dividually. This restriction leads to a conservation of all multipole moments in a ( d − n )-dimensional subspaceorthogonal to these subsystems.Coming back to our two-dimensional lattice example,take a collection of parallel lattice lines that cover thewhole lattice. For concreteness, let us take a collectionof lattice rows that are parallel to ˆ x . Imposing chargeconservation along each row is equivalent to fixing thetotal charge at each position along ˆ y, which is the nor-mal vector to these subsystems. Thus, for any arbitraryfunction f ( y ), the conservation of the quantity Q = (cid:88) x,y f ( y ) q ( x, y ) , (8)where q ( x, y ) is the charge at a site with coordinates( x, y ), is guaranteed by the U (1) subsystem charge con-servation. For example, by taking f ( y ) = y m , we cansee that all multipole moments along ˆ y , such as the P y component of the dipole moment, the Q yy componentof the quadrupole moment, etc., are conserved. Simi-larly, imposing charge conservation along every row ofsites parallel to the ˆ x -axis in a 3D lattice leads to theconservation of all multipole moments in the yz -plane,e.g., P y , P z , Q yy , Q yz , Q zz , etc.Furthermore, we can impose subsystem charge con-servation along two different families of subsystems si-multaneously, e.g., rows and columns in 2D. For a two-dimensional lattice this leads to a conservation of bothcomponents of the dipole moment as well as all highermultipole moments diagonal in either x or y coordinates.However, these subsystem symmetries do not guaranteeconservation of all multipole moments with componentsalong the subsystem. In particular, conservation of theoff-diagonal component of the quadrupole moment Q xy is not achieved by imposing charge conservation alongrows parallel to ˆ x and columns parallel to ˆ y . Instead, onewould have to impose charge conservation along rows ofsites that are perpendicular to the xy -plane. In 2D, forexample, this translates into requiring charge conserva-tion at each individual site of the lattice, which triviallyleads to a conservation of all multipole moments of suchsystem. For the majority of this article we will be fo-cused on the simple case of 1D and 2D systems havingsubsystem symmetries along the rows and/or columns.After that brief discussion let us develop an LSM-typetheorem for systems with subsystem symmetry. First, letus consider a periodic L x × L y rectangular lattice witha Hamiltonian H that conserves U (1) charge along ev-ery row and every column of the lattice, i.e., we have[ H, U ,x ( α )] = [ H, U ,y ( α )] = 0 for every x , y , and α . Particle and dipole conserving Hamiltonians satisfy-ing these criteria can be built from subsystem-symmetricring-exchange type terms .Now, consider the following twist operator acting alonga single column of sites with fixed coordinate x = x : U Y,x = exp (cid:32) πiL y (cid:88) y y ˆ n x ,y (cid:33) . (9)For a periodic lattice and a dipole-conserving Hamilto-nian built from local ring-exchange type interactions (say,of fermions) such as H = − t (cid:3) (cid:88) r (cid:16) c † r +ˆ x c r c † r +ˆ y c r +ˆ x +ˆ y + h.c. (cid:17) , (10)we can calculate the energy difference between theground state | Ψ (cid:105) , and a twisted “variational” state U Y,x | Ψ (cid:105) : (cid:104) Ψ | U − Y,x HU Y,x − H | Ψ (cid:105) = − t (cid:3) (cid:20) (cid:16) e πi/N y − (cid:17) (cid:88) y (cid:104) c † x + a,y c x ,y c † x ,y + a c x + a,y + a (cid:105) + (cid:16) e − πi/N y − (cid:17) (cid:88) y (cid:104) c † x ,y c x − a,y c † x − a,y + a c x ,y + a (cid:105) (cid:21) + h.c. (11)We can follow analogous calculations to Refs. 1 and 7by expanding the exponents in the powers of 1 /N y , andassuming that the ground state preserves at least oneof the reflection symmetries. The constant term in theTaylor expansion of the exponential factors immediatelycancels, but we we need to check that the next order termalso vanishes so that the first non-vanishing term is actu-ally the second-order term from the exponential, which would imply that the energy difference is ultimately ofthe order O (1 /N y ). To see this, after expanding the ex-ponential, consider a pair of plaquettes related by mirrorˆ M y : y → − y and consider the following sum of two oftheir ring-exchange terms from Eq. 11:2 πiN y (cid:104) c † x + a,y c x ,y c † x ,y + a c x + a,y + a (cid:105)− πiN y (cid:104) c † x + a, − y c x , − y c † x , − y − a c x + a, − y − a (cid:105) . (12)We note that the second term, having an opposite sign,comes from the hermitian conjugate part of the overallHamiltonian (10). If the ground state is an eigenstateof the reflection symmetry, ˆ M y | Ψ (cid:105) = ±| Ψ (cid:105) , we canrewrite the second term as: (cid:104) Ψ | c † x + a, − y c x , − y c † x , − y − a c x + a, − y − a | Ψ (cid:105) = (cid:104) Ψ | ˆ M − y c † x + a, − y c x , − y c † x , − y − a c x + a, − y − a ˆ M y | Ψ (cid:105) = (cid:104) Ψ | c † x + a,y c x ,y c † x ,y + a c x + a,y + a | Ψ (cid:105) , (13)and so we see that the sum in Eq. 12 exactly vanishes.The same analysis is applicable to every other pair ofplaquettes that are related by ˆ M y , and so we concludethat the first non-vanishing term in Eq. 11 is of the order O (1 /N y ). Therefore, in the thermodynamic limit, where N y → ∞ , the state | ˜Ψ (cid:105) = U Y,x | Ψ (cid:105) has either exactlythe same energy as the ground state | Ψ (cid:105) , or it is anexcited state with an infinitesimally small energy.Using this result, if we can now show that | Ψ (cid:105) and | ˜Ψ (cid:105) are orthogonal, then the system must necessarilybe gapless (or at least have ground state degeneracy).To this end, let us assume that the ground state doesnot spontaneously break the translational symmetry, so | Ψ (cid:105) will be an eigenstate of the translation operator T x .Using T x | Ψ (cid:105) = e iP x | Ψ (cid:105) it follows that T x | ˜Ψ (cid:105) = T x U Y,x T − x T x | Ψ (cid:105) = e iP x + πiLy (cid:80) Lyy = a y (ˆ n x a,y − ˆ n x ,y ) | ˜Ψ (cid:105) . (14)With the subsystem charge conservation in place, we canintroduce a subsystem polarization associated with a sub-system s as follows: P j s = e π Im log (cid:104) Ψ | U j, s | Ψ (cid:105) (15)where U j, s = exp (cid:32) πiL j (cid:88) r ∈ s x j ˆ n r (cid:33) , (16)and where j = x, y. Now the condition on the groundstate momentum shift to be an integer times 2 π can beunderstood as a condition on the polarizations of twoneighboring subsystems:e πiLy (cid:80) Lyy = a y (ˆ n x a,y − ˆ n x ,y ) | ˜Ψ (cid:105) = | ˜Ψ (cid:105)⇔ e (cid:0) P yx = x + a − P yx = x (cid:1) ∈ Z , (17)where P yx = x is the polarization (15) along the columnwith fixed coordinate x = x . In other words, the ˆ x lat-tice derivative of the y -polarization must be an integer.In general, since we can translate by any number of lat-tice constants in the x -direction, each column must have P yx = x that differ from each other at most by an integer.While this may seem at odds with translation symmetry,the integer ambiguity in the polarization for periodic sys-tems implies that the subsystem polarizations can differby an integer (or more generally can differ by a polariza-tion quantum) and still maintain translation symmetry.Now, recall the statement from the previous subsection,that, in order for dipole-conserving system on an N × N lattice to support a gap, we require the total polarizationto vanish. Combining this result with Eq. 17 we find thatthe polarization of each subsystem can take only the fol-lowing possible of values if the system is gapped: P j s = n j s N e mod e, n j s ∈ Z . (18)Next, we can derive a condition using the fact thatthe ground state | Ψ (cid:105) will also be an eigenstate of thetranslation operator T y : T y | Ψ (cid:105) = e iP y | Ψ (cid:105) . Hence forthe twisted state | ˜Ψ (cid:105) we find: T y | ˜Ψ (cid:105) = T y U Y,x T − y T y | Ψ (cid:105) = e iP y + πiNy (cid:80) Nyy =1 ˆ n x ,y | ˜Ψ (cid:105) . (19)Thus, | ˜Ψ (cid:105) is an eigenstate of T y with eigenvalueexp( iP y + 2 πiν x ) , where ν x is the filling factor of thesubsystem x = x . Hence, the states | ˜Ψ (cid:105) and | Ψ (cid:105) areorthogonal unless each column has integer filling ν x . Sothe system will either have degenerate ground states orgapless excitations if ν x / ∈ Z for any column. We can goback and repeat this analysis for subsystem symmetriesat fixed y . We will find that T x will require that thesubsystem filling factors ν y must all be integers for theground state to be gapped/non-degenerate. Analogously, T y will require that the subsystem polarizations P xy = y must all be the same up to an integer (polarization quan-tum) for the ground state to be gapped/non-degenerate.Although we have chosen a specific Hamiltonian and im-posed reflection symmetry to illustrate that the twistedstate has low energy, we expect that the result is muchmore general , just as the original LSM result is.One consequence of these results is that the regu-lar LSM theorem derived for systems with global U (1)charge conservation symmetry must be satisfied alongevery subsystem individually. In the original case, theLSM theorem states that for the ground state to beunique, the particle number per unit cell calculatedacross the whole lattice must be an integer. Our argu-ments invoking subsystem symmetry and translation canbe straightforwardly applied to d -dimensional Hamilto-nians conserving total U (1) charge across n -dimensionalsubsystems. We can use this logic to formulate a gener-alization of the LSM theorem to classes of Hamiltoniansthat posses not only global U (1) symmetry but also sub-system U (1) symmetries: Consider a d -dimensional periodic lattice with a short-range Hamiltonian that respects U (1) subsystem chargeconservation across a family of parallel n -dimensionalsubspaces. Assume that the ground state does not spon-taneously break translational symmetry along the subsys-tem. For an arbitrary lattice there is at least one low-energy state degenerate, or infinitesimally close in en-ergy, to the ground state, if the particle number per unitcell ν s in any particular n -dimensional subsystem s isnot an integer. Furthermore, the low-energy state hasa crystal momentum in the j -direction (associated to alattice translation operator T j that leaves the subsysteminvariant) differing from the ground state by an amount∆ P j = 2 π V s L j ν s , where V s is the volume of an individualsubsystem, and L j is the lattice size along the ˆ x j .To provide a proof for this general statement, we canuse a suitably adapted version of the argument in Ref. 11.Let us consider a general (bosonic or fermionic) Hamilto-nian H defined on a periodic d -dimensional lattice thatconserves the total U (1) charge across an n -dimensionalsubspace spanned by a collection of n linearly indepen-dent primitive lattice vectors { (cid:126)a , ..., (cid:126)a n } , and their inte-ger linear combinations. We will label this subspace as s . There are a family of subsystems “parallel” to s , butfor now let us focus on this one subsystem. The U (1)symmetry operator for this subsystem is given by U , s ( α ) = exp (cid:32) iα (cid:88) r ∈ s ˆ n r (cid:33) , (20)where r is a lattice vector.The most general lowest-order Hamiltonian that actson s while commuting with U , s ( α ) takes the followingform H = (cid:88) r , r ,j J r , r ,j c † r c r ˆ O j + h.c. (21)where J r , r ,j is a set of coupling constants, r , r ∈ s , and ˆ O j are products of particle creation and annihilationoperators that have no support on s . The ˆ O j may be fur-ther constrained by subsystem symmetries for other sub-spaces s (cid:48) , but we will treat them as arbitrary. If we wantto gauge the subsystem U (1) symmetry it requires theintroduction of an ( n + 1)-dimensional vector-potential A s i , i = 0 ...n associated with subsystem s . Individualterms in the Hamiltonian Eq. (21) couple to the latticegauge field, which modifies the overall Hamiltonian witha subsystem Peierls factor: H ( A s ) = (cid:88) r , r ,j (cid:16) J r , r ,j e iA s r , r c † r c r ˆ O j + h.c. (cid:17) . (22)Now we will invoke the momentum counting argumentfor the subsystem s following the presentation of Ref.11, with the main difference being that the gauge fieldis now associated with subsystem U (1) charge conserva-tion instead of a regular global U (1) symmetry. Let usdefine subsystem magnetic flux quantum insertion oper-ators F s j , j = 1 ...n that adiabatically, over time period T , evolves the gauge field A s between two configurationsthat differ by a large gauge transformation performedalong the direction spanned by the (cid:126)a j -th primitive vec-tor. Adiabaticity of the process is required so that thegap never closes during the time evolution and that, bystarting with a ground state | Ψ (cid:105) , we are guaranteed toend up with a (possibly different) state | ˜Ψ (cid:105) which liesin the ground state subspace, after the time evolution isfinished. Let us define the time evolution of the Hamil-tonian as: H j ( t ) = H (cid:18) A s j = 2 πN j a tT (cid:19) , (23)where N j is the lattice period along the primitive vector (cid:126)a j having lattice constant a , and the subsystem latticegauge field is picked to be uniform in space. The corre-sponding time evolution operator is then given by F s j = T exp (cid:32) − i ˆ T dt H j ( t ) (cid:33) , (24)where T denotes time-ordering. The final Hamiltonian H j ( T ) differs from the initial one H j (0) by a large gaugetransformation, which can be implemented by the follow-ing operator: U X j , s = exp (cid:32) πiL j (cid:88) r ∈ s r j ˆ n r (cid:33) , (25)and one can directly verify that: H j ( T ) = U − X j , s H j (0) U X j , s . (26)We can now define the subsystem flux insertion-removaloperator that leaves the Hamiltonian invariant:˜ F s j ≡ U − X j , s F s j . (27)Importantly, the initial ground state | Ψ (cid:105) , in general,might be different from the state | ˜Ψ (cid:105) ≡ ˜ F s j | Ψ (cid:105) obtainedafter the subsystem flux insertion and removal procedure.To see this explicitly, let us consider the action of thetranslation operator T j on the states | Ψ (cid:105) and | ˜Ψ (cid:105) (wenote that these translations preserve the subsystem s ).Provided that the translational invariance is not sponta-neously broken, we must have: T j | Ψ (cid:105) = e iP j | Ψ (cid:105) (28)where P j is the many-body momentum along (cid:126)a j , andis a good quantum number modulo 2 π . To see how T j acts on | ˜Ψ (cid:105) we first note that, since the subsystem gaugefield evolution implemented by F s j does not break trans-lational invariance, we have:[ F s j , T j ] = 0 . (29) However, T j acts non-trivially on the flux removal uni-tary: T j U X j , s T − j = e − πiNj (cid:80) r ∈ s ˆ n r U X j , s . (30)Therefore, the T j eigenvalue of the final state is: T j | ˜Ψ (cid:105) = F s j T j U X j , s T − j T j | Ψ (cid:105) = exp (cid:18) iP j + 2 πi V s N j ν s (cid:19) | ˜Ψ (cid:105) , (31)where V s is the total number of unit cells inside the sub-system s , and ν s is the filling fraction of said subsystem ν s = N s V s , where N s is the total particle number in s . Wethus see that the momentum along (cid:126)a j of the two states | Ψ (cid:105) and | ˜Ψ (cid:105) differs by:∆ P j = 2 π V s N j ν s . (32)Hence, whenever V s N j ν s is not an integer, the two states | Ψ (cid:105) and | ˜Ψ (cid:105) must be orthogonal to each other.The same argument can be applied to the fluxinsertion-removal procedure for every direction along thesubsystem s . Therefore, for the ground state to beunique, we must require that: V s N j ν s ∈ Z , j = 1 ...n. (33)Since each N j is a divisor of V s the ratios V s /N j areall integers. However, for this set of conditions to besatisfied for arbitrary lattice sizes, we must require ν s itself to be integer. For example, in the case where all N i are co-prime with each other, the only way to satisfy allof the conditions (33) is to require that ν s ∈ Z . (34)We thereby arrive at the theorem stated at the end ofthe Section II B.We can also provide a lower bound on the ground statedegeneracy in the case when ν s / ∈ Z . Assuming that V s N j ν s = p s j q s j , where the pair of integers p s j and q s j are co-prime for all j = 1 ...n : GSD ≥ n (cid:89) j =1 q s j . (35)If we now consider subsystem charge conservation along m subsystems s i , i = 1 , . . . m, which are all translation-ally invariant in the j -th direction (e.g., parallel rows), wecan combine different twist operators U X j , s i , i = 1 , ..., m to generate low-lying states where the many-body mo-mentum is shifted when compared to the ground stateby: ∆ P j = 2 π V s N j m (cid:88) i =1 n i ν s i , (36)where n i are integers and V s is the total number of unitcells in every subsystem. The total number of inequiv-alent values (mod 2 π ) that ∆ P j can take is the leastcommon multiple of corresponding q s i j integers. Thus,the ground state degeneracy associated with translations T j is bounded below by GSD j ≥ lcm (cid:0) q s j , q s j , ..., q s m j (cid:1) . (37)Hence, the total ground state degeneracy for a d -dimensional lattice with m subsystem symmetries isbounded below by: GSD ≥ d (cid:89) j =1 lcm (cid:0) q s j , q s j , ..., q s m j (cid:1) (38)where we set q s i j = 1 if a particular subsystem s i is notleft invariant under the action of T j .To give an example, imagine a two-dimensional latticewhere, on top of the regular global U (1) symmetry weimpose independent subsystem U (1) symmetries alongtwo rows r and r defined by the equations y = y and y = y respectively. Both r and r are invariant underthe action of T x operator. As an example, let us nowchoose the fillings of these two rows to satisfy: V r N x ν r = 12 ≡ q r x , V r N x ν r = 14 ≡ q r x . (39)Applying the unitary operator U X, r , we can concludethat there are at least two states in the ground statesubspace which are eigenstates of the translation oper-ator T x with eigenvalues e iP x and e iP x + iπ . Similarly,applying unitary U X, r we find at least four translation-ally invariant states with eigenvalues e iP x , e iP x + iπ/ ,e iP x + iπ , and e iP x + i π/ . Since two of these values havealready appeared when we used U X, r , and we cannoteasily distinguish two twisted states with the same mo-mentum, we conclude that the total ground state degen-eracy is at least 4 = lcm( q r x , q r x ).As another example let us consider a pair of subsystemsymmetries imposed along two columns c and c whichare defined by the equations x = x and x = x withfillings V c N y ν c = 12 ≡ q c y , V c N y ν c = 13 ≡ q c y . (40)Analogous to the previous paragraph, these subsys-tems are invariant under T y , and acting on the groundstate with unitary operators U Y, c and U Y, c generateslow-lying states with translation eigenvalues e iP y ande iP y + iπ for the first operator, and e iP y , e iP y + i π/ , ande iP y + i π/ for the second one. Additionally, we cancombine U Y, c with U Y, c to obtain a low-energy statewith the translation eigenvalue equal to e iP y + iπ/ ande iP y + i π/ . Thus, we conclude that the total numberof low-lying states that can be generated by the columntwist operators is 6 = lcm( q c y , q c y ). For a two-dimensional system that is translationally in-variant simultaneously along both ˆ x and ˆ y , we expect thefillings of all rows to be the same if the ground state doesnot spontaneously break translation symmetry, therefore q x = q r x = q r x = ... , and so the ground state degeneracyassociated with translations T x is GSD x ≥ lcm ( q r x , q r x , ... ) = q x . (41)Similarly, the fillings of all columns must also be the samegiving us q y = q c y = q c y = ... , leading to the ground statedegeneracy associated with translations along ˆ y to be GSD y ≥ lcm (cid:0) q c y , q c y , ... (cid:1) = q y . (42)The total ground state degeneracy of such system is thenbounded from below by the product of the two factors: GSD ≥ q x q y . (43) III. APPLICATION: PLATEAUS INMAGNETIZATION AND MAGNETIZATIONGRADIENTS
Let us take these concepts and apply them to spin sys-tems with an aim toward making physical predictions. Inthe work of Oshikawa, Yamanaka, and Affleck , bosonicspin counterparts of the twist operators (1) were suc-cessfully used to derive conditions for the appearance ofmagnetization plateaus as a function of applied externalmagnetic field in, e.g., spin chains. Here we will definebosonic spin counterparts of the multipole twist oper-ators Eqs. (2), (9) to derive conditions in spin laddersystems for the appearance of plateaus of the gradient ofmagnetization as a function of an applied magnetic field gradient placed across the ladder. Explicitly, we imag-ine tuning the magnetic field so that the system is ona conventional magnetization plateau, and then test howthe gradient of the magnetization responds to a magneticfield gradient around a uniform background field. We willwant to distinguish between two cases: (i) the system hasa non-constant magnetization gradient response, or (ii)the system exhibits a plateau in the magnetization gra-dient. These physical phenomena are closely related tothe recent work on dipole insulators . In the language ofRef. 18, since the system is tuned to a conventional mag-netization plateau, the analogous charge system would bea charge insulator. However, case (i) would be a chargeinsulator but a dipole metal, while (ii) would be both acharge insulator and a dipole insulator.To illustrate these two possibilities we will consider sys-tems with axial spin rotation symmetry along subspaces,i.e., models with U (1) subsystem symmetry correspond-ing to a conservation of the total S z along each subspace.A unitary operator corresponding to a U (1) subsystemsymmetry associated to a particular subsystem s reads: U , s ( α ) = exp (cid:32) iα (cid:88) r ∈ s ˆ S z r (cid:33) . (44)This operator rotates all spins belonging to s around the z -axis by the same amount. The corresponding conservedquantity is the total S z magnetization on s . A. One-dimensional Spin Ladder Model
As an explicit test system let us first consider a two-legspin- S ladder that is stretched along the ˆ x -axis with peri-odic boundary conditions in the x -direction. We will alsoenforce two U (1) subsystem symmetries, one of whichimplies conservation of the total magnetization on thetop leg (which we label with ‘ ↑ ’), and the other whichimplies conservation of the total magnetization of thebottom leg (which we label with ‘ ↓ ’). Let us assumethat the system’s ground state does not break transla-tional symmetry, and has a fixed total magnetizationfor some range of values of an external magnetic field h < B z < h , i.e., the state of the system is at a magne-tization plateau. Applying the conventional magnetiza-tion plateau argument to a two-leg ladder spin systemwe find the magnetization per spin in a two-leg ladder ofthe size L x × L y = N x a × a : M z ≡ N x N x (cid:88) x =1 ( S zx, ↑ + S zx, ↓ ) (45)takes half-integer values, i.e., M z = 0 , ± , ± , etc.To preserve the subsystem symmetries we can build aHamiltonian from spin ring-exchange terms, e.g., nearestneighbor ring exchanges: H = J (cid:3) (cid:88) x (cid:16) S + x, ↑ S − x, ↓ S + x +1 , ↓ S − x +1 , ↑ + h.c. (cid:17) . (46)Similar to the ring-exchange model studied in the previ-ous section, where such terms tunneled charge dipoles,here they can be interpreted as tunneling terms for mag-netic quadrupole moments (spins are already magneticdipoles so separating opposite spins by a distance to cre-ate a “dipole of spins” creates a magnetic quadrupole).Now, consider the following unitary twist operator actingalong one of the legs of the ladder: U X, ↑ = exp (cid:32) πiL x (cid:88) x xS zx, ↑ (cid:33) . (47)Under the action of the operator U X, ↑ each term in H ismodified as: U − X, ↑ S + x, ↑ S − x, ↓ S + x +1 , ↓ S − x +1 , ↑ U X, ↑ = e πiNx S + x, ↑ S − x, ↓ S + x +1 , ↓ S − x +1 , ↑ . (48)Therefore, we can show that for the ground state | Ψ (cid:105) , which we assume preserves translation and reflectionsymmetry along ˆ x, we have: (cid:104) Ψ | U − X, ↑ HU X, ↑ − H | Ψ (cid:105) = O (cid:18) N x (cid:19) . (49) And so, similar to the previous section, we see that in thethermodynamic limit N x → ∞ the state | ˜Ψ (cid:105) = U X, ↑ | Ψ (cid:105) lies in, or infinitesimally near, the ground state subspace.Now let us check if | ˜Ψ (cid:105) = U X, ↑ | Ψ (cid:105) is orthogonal to | Ψ (cid:105) . Following logic that should now be apparent, we cancompute the commutation relation between the transla-tion operator and U X, ↑ to find: T x U X, ↑ T − x = U X, ↑ e πiS z , ↑ − πiNx (cid:80) Nxx =1 S zx, ↑ . (50)Therefore, starting from a ground state | Ψ (cid:105) having awell-defined many-body momentum T x | Ψ (cid:105) = e iP x | Ψ (cid:105) ,we find the state | ˜Ψ (cid:105) has the eigenvalue: T x | ˜Ψ (cid:105) = T x U X, ↑ T − x T x | Ψ (cid:105) = e iP x +2 πiS z , ↑ − πiNx (cid:80) Nxx =1 S zx, ↑ | ˜Ψ (cid:105) . (51)A notable difference from the previous section is the ap-pearance of the extra term 2 πS , ↑ in the phase factorwhich can be integer or half-integer depending on thespin model of interest. From this analysis we concludethat the two states are orthogonal unless S ↑ − m ↑ ∈ Z , where m ↑ = N x (cid:80) x S zx, ↑ . We can obtain a similar condi-tion by considering the unitary operator U X, ↓ that actson the bottom leg. Thus, the ground state can be uniqueonly if the spin minus the average magnetization m z ↑ / ↓ ofthe top row or bottom row of spins are both integers: S ↑ / ↓ − m z ↑ / ↓ ∈ Z . (52)Let us analyze these conditions in more detail. We canactually make a more physically intuitive statement bynoticing that the sum of average magnetizations of bothrows must be an integer as well: m z ↑ + m z ↓ = 2 M z ∈ Z , (53)where we have used the assumption that our system istuned to a conventional magnetization plateau, and thefact that the magnetization per spin (Eq. 45) must be amultiple of 1 / m z ↓ =2 M z − m z ↑ , m z ↑ = n − S, n ∈ Z . (54)Combining these statements we end up with the followingcondition for the magnetization gradient in the directiontransverse to the legs of the ladder:∆ y m z ≡ ( m z ↑ − m z ↓ ) = 2( n − S − M z ) . (55)In this equation the spin S (in a unit cell on a singleleg) is fixed, and since we are tuned to a magnetizationplateau M z is a multiple of 1 / . Thus, we expect themagnetization gradient to have plateaus at only even oronly odd integer values where the parity is determinedby whether the sum of the total spin S and magneti-zation M z is integer, or half-integer respectively. Al-ternatively, since the total magnetization of the ladderis vanishing, we can recast the magnetization gradientas a magnetic quadrupole moment Q Myz where z -orientedmagnetizations are separated along the y -direction. Ourresults imply that the system has a plateau of Q Myz as afunction of magnetic field gradient. If the total magne-tization were on a plateau, but non-vanishing then theconversion to a magnetic quadrupole moment would de-pend on our choice of coordinate origin. In all of theexamples here the total magnetization vanishes so thisissue does not arise.We corroborate our results using the numerics pre-sented in Fig. 1. In this figure we compare the magneticresponses of two types of spin-1/2 ladders. In Fig. 1a,cwe show results for a two-leg spin-ladder with nearestneighbor XY couplings, while in Fig. 1b,d we show re-sults for a two-leg spin ladder with ring-exchange terms(c.f. Eq. 46). For each system we first show their re-sponse to a uniform magnetic field. For the XY ladder(Fig. 1a) we see magnetization plateaus at M z = 0 , ± / S − M z ) ∈ Z is the condition for a plateau for a two-leg system. Wealso find that the ring exchange model exhibits magne-tization plateaus at M z = 0 , ± / S − M z ) ∈ Z .In Fig. 1a we also overlayed a dashed red line showingthe magnetization response of a single, nearest-neighborspin chain coupled via (2 S xi S xi +1 + S yi S yi +1 ) interactions.These interactions explicitly break the axial U (1) symme-try corresponding to spin rotations around the ˆ z axis, andhence the system does not exhibit discrete magnetizationplateaus, but instead smoothly interpolates between thetwo configurations where the average magnetization sat-urates. We chose to compare the XY ladder with an XYspin chain having broken U (1) spin symmetry to make ananalogy to the comparison between Figs. 1c,d betweenthe XY ladder and ring-exchange ladder, where the for-mer has broken U (1) subsystem symmetry.Now we want to examine the magnetization gradientresponse of the XY ladder and the ring-exchange ladder.Let us consider applying a magnetization gradient cen-tered around zero uniform applied magnetic field, i.e.,we apply a B z = + h to the top leg of the ladder and B z = − h to the bottom one. The total magnetiza-tion of both systems stays at a magnetization plateauwith M z = 0. For the XY ladder that respects only aglobal axial U (1) symmetry, but not a subsystem U (1),we see a smooth interpolation of the magnetization gra-dient between the saturation points at ∆ y m z = − y m z = +1. This is quite similar to the behavior of the magnetization of the XY spin chain shown in Fig. 1athat does not respect the global axial U (1) symmetry.For the ring exchange model, which respects axial sub-system U (1) symmetry, we find that ∆ y m z exhibits a se-ries of plateaus. The two most stable ones are located ex-actly at ∆ y m z = ± y m z = 0,however, as we show in the inset plot in Fig. 1d, this - - - M z B z -
12 -4 -2 0 2 4 (a) - - M z B z -
12 -4 0 4 (b) - - ∆ y m z ∆ y B z -101 -2 0 2 4 (c) - - ∆ y m z ∆ y B z -101 -5 0 5 10 (d) p l a t e a u w i d t h FIG. 1. Magnetization (a, b) and magnetization gradient (c, d) responses of a two-leg spin-1/2 ladder to an appliedexternal magnetic field B z and magnetic field gradient ∆ y B z respectively. In plots (a) and (c) , the two-leg ladder is cou-pled via spin-anisotropic XY interactions that do not preservesubsystem symmetry, while the ladder in plots (b) and (d) is coupled via ring-exchange interactions that have subsystemsymmetry. Additionally, for we include the magnetization re-sponse to a constant B z of a chain that does not respect global U (1) S z rotation symmetry, which is depicted as a dashed redline in (a) . We superimposed numerical data for ladders with4, 6, 8, 10, and 12 rungs. We clearly see that both magnetiza-tion and its gradient experience plateaus for the ring-exchangeHamiltonian, while there are no plateaus of magnetizationgradient in the data for the XY-coupled ladder. In the mag-netization gradient data for the ring-exchange model we see asmall plateau at ∆ y m z = 0 that monotonically shrinks withincreasing system size as shown by the inset plot. It is pos-sible that this plateau will not survive the thermodynamiclimit. plateau is shrinking rapidly as we increase the systemsize, and it is not clear if it will survive or not in thethermodynamic limit.
1. Ising-Coupled Spin Ladder
As a brief aside, we can further illustrate the physics ofmagnetization gradient plateaus in one-dimensional lad-ders by connecting them to the ordinary magnetizationplateaus in an effective single spin chain. To see this,consider a spin-1/2 ladder of length L x in a magneticfield with ring-exchange, Ising, and Zeeman couplings: H = (cid:88) x [( J (cid:3) S + x, ↑ S − x +1 , ↑ S + x +1 , ↓ S − x, ↓ + h.c. ) + λS zx, ↑ S zx, ↓ + h ↑ S zx, ↑ + h ↓ S zx, ↓ ] , (56)where S x, ↑ / ↓ is the spin − / x and thetop ( ↑ ) or bottom ( ↓ ) leg. This Hamiltonian commuteswith the total magnetization operator of the entire lad-der ( (cid:80) x [ S x, ↑ + S x, ↓ ]), and the individual magnetization0operators of each leg ( (cid:80) x S x, ↑ and (cid:80) x S x, ↓ ). The formeris a U (1) global symmetry, while the latter are a pair of U (1) subsystem symmetries.Here, we will be interested in the limit λ (cid:29) J (cid:3) , | h ↑ / ↓ | ,where every rung on the ladder will be pinned such that (cid:104) S zx, ↑ S zx, ↓ (cid:105) = − . In this limit, the total magnetizationof the system is fixed to be an integer M z = 0, and thesystem is on a magnetization plateau. There are twoconfigurations that satisfy this constraint: (cid:104) S zx, ↑ (cid:105) = ± , (cid:104) S zx, ↓ (cid:105) = ∓ . Let us define a new effective spin degree offreedom on each rung, ˜ S r such that ˜ S zx = S zx, ↑ = − S zx, ↓ .Using this, we can also define: ˜ S + x = S + x, ↑ S − x +1 , ↑ , ˜ S − x = S − x, ↓ S + x +1 , ↓ . Combined with ˜ S zx , these operators satisfythe spin-1/2 algebra, and the spin ladder becomes: H = J (cid:3) (cid:88) x ˜ S + x ˜ S − x +1 + ( h ↑ − h ↓ ) ˜ S zx , (57)which is the Hamiltonian for a single XY spin chain inan effective magnetic field given by h ↑ − h ↓ .Now let us consider the response of this system to ex-ternal magnetic fields. When this system is placed in auniform physical magnetic field ( h = h ), the effectivemagnetic field vanishes so the system does not develop amagnetization as might be expected. However, if we in-stead consider a physical magnetic field gradient parallelto the ladder rungs (e.g., h = − h ), the effective mag-netic field is non-vanishing and the system can develop aneffective magnetization. The key point is that the mag-netization of the effective spins in Eq. 57 is equal to halfthe magnetization gradient of the original spin ladder:1 L x (cid:88) x ˜ S zx = 12 1 L x (cid:88) x [ S zx, − S zx, ] . (58)In conclusion, the effective magnetic field and magneti-zation associated to Eq. 57 are respectively the physi-cal magnetic field gradient and magnetization gradientof the spin ladder Eq. 56. A magnetization plateau forthe effective spins in Eq. 57, is thereby equivalent to amagnetization gradient plateau for the physical spin inEq. 56.It is well known that the XY spin chain has magneti-zation plateaus when the magnetization is equal to ±
12 1 .From this, we can conclude that that the spin ladder Eq.56 is at a plateau in the gradient of its magnetizationwhen ∆ y m z = (cid:80) x ( S zx, − S zx, ) /L x = 2 (cid:80) x ˜ S zx /L x = ± M z = 0, this result agrees with Eq. 55. Itis worth noting that this result is only true in the limit λ (cid:29) | h ↑ / ↓ | . In the opposite limit, | h ↑ / ↓ | (cid:29) λ, J , thesystem will be at ordinary magnetic plateaus where thespins are aligned parallel to the magnetic field. B. Two-dimensional Spin Systems
Now let us consider two-dimensional spin models. Wewill first derive the spin analog of the dipole LSM the- orem from Sec. II A. Working on a square periodic L × L = N a × N a lattice, we will consider spin Hamilto-nians which possess global U (1) symmetry that acts byrotating all spins around the ˆ z -axis by the same amount.The corresponding conserved quantity is the total mag-netization M z of the system. We will additionally im-pose conservation of two components of the magneticquadrupole moment Q Mxz and Q Myz which is the analog ofthe conservation of the x and y components of the dipolemoment for particles whose charge under the global U (1)symmetry is itself a magnetic dipole moment pointing inthe z -direction.This setup is entirely analogous to the one consideredin Sec. II A. It is natural then to consider Hamiltoni-ans where the lowest-order dynamical terms are built ofbosonic spin ring-exchange terms, as in (46). Such sys-tems were recently discussed in the literature whereit was shown, that they naturally couple to the back-ground symmetric rank-2 gauge field A xy with a Peierlsphase factor. To derive an LSM-type theorem we willbriefly recount the argument already discussed in de-tail in the context of dipole-conserving systems in Sec.II A. We start by adiabatically driving the value of thebackground field A xy from 0 to 2 π/N a over time pe-riod T . This evolves the ground state of the system from | Ψ(0) (cid:105) to | Ψ( T ) (cid:105) . As this process is performed uniformlyacross the lattice, without breaking translational sym-metry, | Ψ( T ) (cid:105) will remain an eigenstate of T x and T y ,provided that the translational invariance of the initialHamiltonian was not spontaneously broken in its groundstate | Ψ(0) (cid:105) . Then, we apply the unitary twist operator U XY = exp (cid:32) πiaL (cid:88) r xy S z r (cid:33) (59)which removes the change in A xy and brings the Hamil-tonian back to its original form. The resulting eigen-state U − XY | Ψ( T ) (cid:105) has an energy infinitesimally close tothe ground state in the thermodynamic limit, and it maybe different from the original ground state | Ψ(0) (cid:105) . If weconsider the commutation relation between translationsin the ˆ x -direction and the twist operator, we obtain anadditional phase factor: T x U − XY T − x = U − XY e πiL (cid:80) r yS z r e − πi (cid:80) Ny =1 yS zx =1 ,y (60)where the first extra factor on the RHS of the equationcontains the total magnetic quadrupole moment Q M,totyz of the system in the exponential. The phase in the secondfactor can take either integer or half-integer multiples of2 π depending on whether the spin S is integer or half-integer, and whether the value of N ( N + 1) / T x tobe the same for U − XY | Ψ( T ) (cid:105) and | Ψ(0) (cid:105) the extra phasefactors appearing in (60) must be trivial. This yieldsthe following condition for the uniqueness of the groundstate: N ( N + 1)2 S + (cid:88) r yS z r N a ∈ Z , (61)1which is very similar to the condition obtained in Sec.II A. For instance, consider integer S . The condition (61)then requires that, for the ground state to be unique,the total magnetic quadrupole moment Q Myz must be aninteger. We can repeat this calculation using translationin the ˆ y -direction to derive N ( N + 1)2 S + (cid:88) r xS z r N a ∈ Z , (62)which gives a similar condition but for Q Mxz . Now, let us move on to spin systems with U (1) subsys-tem symmetry where the S z spin component is conservedon rows and columns of the lattice. We can again con-sider a twist operator that acts along a single column ofspins with fixed coordinate x = x : U Y,x = exp (cid:32) πiL y (cid:88) y yS zx ,y (cid:33) . (63)For a periodic lattice and a subsystem symmetric Hamil-tonian built from local ring-exchange terms such as Eq.(46), we can compare the energy of a twisted state withthe original ground state | Ψ (cid:105) . If we assume that | Ψ (cid:105) does not spontaneously break the translational invari-ance and preserves a reflection symmetry ˆ M y : y → − y ,similar to (11) we have: (cid:104) Ψ | U − Y,x HU Y,x − H | Ψ (cid:105) = O (cid:18) N y (cid:19) . (64)Therefore, in the thermodynamic limit, where N y → ∞ ,the state | ˜Ψ (cid:105) = U Y,x | Ψ (cid:105) has either exactly the sameenergy as the ground state | Ψ (cid:105) , or it is an excited statewith an energy infinitesimally close to the ground state.We now want to see if | Ψ (cid:105) and | ˜Ψ (cid:105) are orthogo-nal. Assuming that the ground state does not sponta-neously break the translational symmetry, i.e., T x | Ψ (cid:105) =e iP x | Ψ (cid:105) , we can show that the translation eigenvaluefor | ˜Ψ (cid:105) may take a dstinct value: T x | ˜Ψ (cid:105) = T x U Y,x T − x T x | Ψ (cid:105) = e iP x + πiLy (cid:80) y y ( S zx ,y − S zx ,y ) | ˜Ψ (cid:105) . (65)Similar to the subsystem polarization introduced in Sec.II A, we introduce an analogous notion for spin systems– a subsystem quadrupole polarization: Q Mjz ( s ) = 12 π Im log (cid:104) Ψ | U j, s | Ψ (cid:105) , (66)where U j, s = exp (cid:32) πiL j (cid:88) r ∈ s x j S z r (cid:33) , (67)and where j = x, y . Therefore, for the ground stateto be unique we must have that the pair of magnetic quadrupole moments Q Myz computed along neighboringsubsystems must differ by an integer number:e πiLy (cid:80) y y ( S zx ,y − S zx ,y ) | ˜Ψ (cid:105) = | ˜Ψ (cid:105)⇔ Q Myz ( x = x + 1) − Q Myz ( x = x ) ∈ Z , (68)where Q Myz ( x = x ) is the magnetic quadrupolar polariza-tion (66) along the column with fixed coordinate x = x .Therefore, for the states | Ψ (cid:105) and | ˜Ψ (cid:105) to have the sameeigenvalue of the translation operator T x we need to re-quire that the difference between the subsystem magnetic Q Myz quadrupole moments computed along two the adja-cent rows of spins is an integer number. In general, sincewe can translate by any number of lattice constants inthe x -direction, each column must have Q Myz that differat most by an integer if we want to preserve translationsymmetry and have a unique ground state. Noting thatthe total magnetic quadrupole moment Q Myz should sat-isfy the Eq. 62 we can add in the relationship betweensubsystem quadrupole moments (68) to see that on a N × N lattice subsystem quadrupolarization must takethe following set of values: Q Myz ( x = x ) = nN − N + 12 S, n ∈ Z (69)and similarly for Q Mxz ( y = y ) on every column with fixedcoordinate y = y .Now, considering translations along ˆ y we find: T y | ˜Ψ (cid:105) = e iP y +2 πiS zx , − πiNy (cid:80) Nyy =1 S zx ,y | ˜Ψ (cid:105) , (70)which means that for the ground state | Ψ (cid:105) to be uniquewe need the average magnetization m zx = x of a single col-umn at x = x to satisfy: ( S − m zx = x ) ∈ Z , with S beingthe total spin per unit cell of a subsystem. Hence, thestates | Ψ (cid:105) and | ˜Ψ (cid:105) are orthogonal unless the averagemagnetization of each subsystem s satisfies S − m z s ∈ Z . (71)The physical consequences of these results are moresubtle than the ladder case. We have found that in or-der for a system with magnetic quadrupole conservationto have a unique ground state the spin and magneticquadrupolarization must satisfy an integer constraint.Furthermore, if the system has subsystem spin-rotationsymmetry then each subsystem has to be on a magneti-zation plateau for the ground state to be unique. Thus,in the latter case, if we apply a spatially varying mag-netic field that is constant along a family of subsys-tems, and weak enough not to drive any subsystem off itsplateau, then the system will have a constant magneti-zation plateau response even to a spatially varying mag-netic field. For example, if we have subsystem symmetryin 2D along rows and columns parallel to x and y respec-tively, then our system will have a non-varying responseto magnetic fields having only x or only y dependence aslong as the field applied to any given subsystem is not2strong enough to drive it off its plateau. In the formercase without subsystem symmetry the system can exhibita plateau in the magnetic quadrupolarization in the pres-ence of a pure magnetic field gradient, i.e., a non-uniformmagnetic field configuration that can have, at most, lin-ear dependence on the spatial coordinates. Both of thesepossibilities suggest that for magnetic systems tuned tomagnetization plateaus there can be a refinement of themagnetic response characterization based on how the sys-tem responds to non-uniform fields. Indeed, the systemswe studied here can exhibit additional types of magneticresponse plateaus when they have unique ground states. IV. LUTTINGER-LIKE THEOREM FORDIPOLES
The LSM theorem has been used in non-perturbativearguments supporting Luttinger’s theorem . At theheart of these arguments is the connection between themomentum of a low-energy excitation and the particlefilling. For a Fermi-liquid this relates the Fermi-surface,where low-energy excitations are created (having momen-tum of order k F ), to the electron filling, even in an in-teracting system. In this section we will apply similararguments to show that some systems having U (1) par-ticle number and dipole conservation can support low-energy excitations having momentum determined by thefilling of dipoles. Here we will just provide an example,and leave a full discussion, and generalization to higherdimensions to future work.Let us consider a two-leg fermion ladder model parallelto the x -direction (let the lattice constant a = 1). Wewill use the Hamiltonian H = J N (cid:88) i =1 ( d † i d i +1 + h.c. ) + U N (cid:88) i =1 n i ↑ n i ↓ , (72)where ↑ / ↓ label the two legs of the ladder, d i ≡ c † i ↓ c i ↑ is a dipole annihilation operator for a dipole parallel to y , i.e., along the rungs of the ladder, c † i ↓ / ↑ is a fermioncreation operator on the lower/upper legs respectively atsite i , and the n i ↓ / ↑ are the fermion density operatorson each leg. In Ref. 18, it was shown that when thissystem is at half-filling ( N F = N ), and U (cid:29) J, thenthe dipole operators effectively become hard-core bosonshaving onsite anticommutation relations { d † i , d i } = 1 , { d † i , d † i } = { d i , d i } = 0 , (73)and commuting off site. Thus, in this limit this modelbecomes a hopping model for y -oriented dipoles that be-have as hardcore bosons. We can identify up-dipoles(down-dipoles) with a configuration where, at a particu-lar unit cell i there is a fermion on the upper leg (lowerleg) and no fermion on the lower leg (upper leg). Basedon this, we can define the total dipole number operatoras N D = (cid:80) Ni =1 [ n i ↑ − n i ↓ ], and the y component of the polarization as p y = N D /N . It is clear that the Hamil-tonian in Eq. 72 conserves the dipole number N D . Sincethe total fermion number ( N F = (cid:80) Ni =1 [ n i ↑ + n i ↓ ]) is alsoconserved, the fermion number on each leg of the lad-der ( N ↑ = (cid:80) Ni =1 n i ↑ and N ↓ = (cid:80) Ni =1 n i ↓ ) is conserved aswell.To prove a Luttinger-like theorem we want to showthat the low-energy modes of this model at some fillingof dipoles have momentum related to that dipole filling.Let us take the ground state of the system to be | Ψ (cid:105) .We will consider the twisted variational state | ˜Ψ (cid:105) =exp(2 πi (cid:80) Nj =1 jn j, ↑ /N ) | Ψ (cid:105) . A calculation analogous towhat we have presented in detail above for the ring ex-change model shows that the energy of this state is within O (1 /N ) of the ground state energy. We can now calculatethe momentum of this state. If | Ψ (cid:105) is an eigenstate of thelattice translation operator T x with eigenvalue e iP x , then | ˜Ψ (cid:105) will have an eigenvalue e iP x +2 πi (cid:80) Ni =1 n i, ↑ /N . Usingthe relation (cid:80) Ni =1 n i, ↑ /N = N ( N D + N F ), (and also that N F = N since the fermions are half-filled), there mustbe a low energy mode with momentum [ P x + π ( p y + 1)],where we recall p y is the y -component of the charge po-larization. Similarly, if we twist the ground state withthe inverse of the operator above we will find anotherlow energy mode with momentum [ P x − π ( p y + 1)]. Werecall that these modes are only guaranteed to be orthog-onal to the untwisted ground state if the polarization p y is not an integer. We will argue below that these pointsform an analog of a Fermi surface for dipoles with Fermiwavevector k ( dipole ) F = π ( p y + 1) . (74)Alternatively, we can derive these results with an ex-plicit solution of this model. If the dipoles are effectivelyhard-core bosons, this model can be transformed into aspin-1/2 XY model using: S αi = 12 (cid:126)c † i σ α (cid:126)c i , where (cid:126)c i = ( c i, ↑ , c i ↓ ) T , (75)so that d † i = 2 S + i , d i = 2 S − i . (76)The resulting spin Hamiltonian is H = 2 J N (cid:88) i =1 (cid:0) S + i S − i +1 + S − i S + i +1 (cid:1) . (77)It is well-known that such an XY model is exactly solv-able in 1D via a Jordan-Wigner transformation: S + i = e iπ (cid:80) i − j =1 f † j f j f † i , S − i = e − iπ (cid:80) i − j =1 f † j f j f i , (78)and the resulting transformed Hamiltonian is: H = 2 J (cid:32) N − (cid:88) i =1 f † i f i +1 + e iπ (cid:80) Nj =1 f † j f j f † N f (cid:33) + h.c. (79)3where f i is the annihilation operator for a Jordan-Wignerfermion on site i. Using these mappings we can identify the low-energyexcitations of the dipole model with the Fermi-surfaceexcitations of the Jordan-Wigner fermions. These exci-tations occur at momentum ± k ( dipole ) F which is directlyproportional to the density of Jordan-Wigner fermions,and through the mappings above to the density of y -dipoles. Precisely we have 2 k ( dipole ) F = 2 πν , where ν isthe fraction of up-dipoles in the system. Alternativelywe can rewrite dipole density as: ν = N ↑ N ↑ + N ↓ = 12 N ↑ − N ↓ N ↑ + N ↓ + 12 = 12 ( p y + 1) , (80)where p y is the charge polarization in the y -direction.Thus we can relate the area enclosed by a Fermi sur-face in 1D to the polarization of the dipole chain in thetransverse direction and we recover Eq. 74.We expect that results like this can apply beyond one-dimensional ladders. As an example, we could considera model for a 2D dipole metal recently discussed in Ref.18. This model is built by stacking dipole ladder mod-els (which are parallel to ˆ x ) into the y -direction andintroducing dipole hopping terms between the nearest-neighbor rungs of two neighboring ladders. Effectively,the model describes a system of free y -dipoles that canmove across a rectangular lattice. It was shown that thismodel can be Jordan-Wigner transformed to a fermionictight-binding model which has a well-defined Fermi sur-face. This transformation translates number operatorsfor y -dipoles into ordinary number operators for theJordan-Wigner fermions. Hence Luttinger’s theorem fora two-dimensional fermionic model, when translated to adipole language, once again relates an area enclosed bya Fermi surface to the density of y -dipoles in the lattice,or, in other words, to the ˆ y polarization of the groundstate.We note a possible connection to the recent work inRef. 34, where elementary dipole particles having a fixeddipole moment were considered. In comparison, however,the statistics of those particles was taken to be fermionic(in our case they are hard-core bosons), and the interac-tions between particles were taken into account. It wasthen argued that this system develops a stable interactingFermi liquid with a Fermi surface elongated in the direc-tion of the dipole moment. A Luttinger theorem for suchfermionic dipoles, which we do not prove here, would alsonecessarily relate an area enclosed by a Fermi surface tothe density of fermionic dipoles, i.e., the polarization ofthe system in the dipole moment direction. V. CONCLUSIONS
In this paper we derived several non-perturbative re-sults for dipole-conserving Hamiltonians and their spincounterparts. We provided a generalization of the LSM theorem to multipole-conserving systems, and find thatfor dipole conserving systems, a unique gapped, transla-tionally invariant ground state is possible only if the bulkpolarization is integer (integer filling of dipoles). A ra-tional polarization of p/q implies that there are at least q degenerate ground states. Furthermore, if the systemboth conserves polarization and has a U (1) charge con-servation symmetry along subsystems, a unique gapped,translationally invariant ground state is possible only ifthe filling in each subystem is an integer. A rationalfilling implies either a gapless system or a ground statedegeneracy. We also provided the spin counterpart ofthis theorem, that applies to spin systems that have con-served magnetic quadrupole moments and possibly pre-serve spin-rotation symmetry on subsystems. These sys-tems can experience plateaus in the magnetic responsein some types of non-uniform fields. Finally we have alsodiscussed a possible extension of a Luttinger-like theoremto dipole systems.From these results, we have been able to place strongconstraints on the low energy physics of systems havingconserved multipole moments. As with the famous re-sults of Lieb, Schultz and Mattis, these results can beused to study strongly correlated systems, where nor-mal perturbative methods fail. Much is still unknownabout multipole conserving systems on lattices , andin the continuum , and our results may prove use-ful in these contexts. These results also hint at possi-ble exotic gapped phases that have fractional polariza-tion/dipole moment, in analogy to topologically orderedsystems having fractional charge. The new types of mag-netic response plateaus we predicted may also belongto topological phases, analogous to the Haldane phasein SPT spin chains. Experimentally, our results can betested in cold-atom systems, where dipole conserving sys-tems can be constructed . These cold-atom systemsmay be an interesting place to look for the aforemen-tioned exotic phases. While we have focused primarilyon 1D and 2D we expect the results can be extendedstraightforwardly to higher dimensions, and with a va-riety of conserved types of multipole moments. Finally,it could prove useful exploring possible connections be-tween our LSM-type theorems and similar results re-cently acquired in the context of systems with higher-form symmetries . Note:
During the preparation of this manuscript webecame aware of a recent work titled “Lieb-Schultz-Mattis type constraints on Fractonic Matter” by HuanHe, Yizhi You, and Abhinav Prem . Our work has someoverlapping concepts and results with this article, butboth were carried out independently. ACKNOWLEDGMENTS
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